Limit Cycle Prediction for a Multi-Degree-of-Freedom System with a Freeplay Non-Linearity

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This paper presents a method for predicting limit cycles of a multi-degree-of-freedom system possessing a freeplay non-linearity. The approach taken is through casting the problem as an integro-differential equation. The method is a development of that previously reported in the literature for the simpler case of such a system with a cubic hardening non-linearity. The system considered is based on aeroelastic applications where structural non-linearities of this kind are encountered. Limit cycles stability is determined using an implementation of Floquet analysis based on extending the Hill's Determinant approach that may be used in analysing the Mathieu equation. The limit cycle predictions and Floquet multipliers are compared against predictions from numerical integration to show the validity of the method. Fast Fourier transform analysis is used to provide comparisons with the predictions of harmonic components from the analytical results. As the Floquet analysis also produces an approximation to the motion of the system in the neighbourhood of a limit cycle, in the case of an unstable limit cycle, it was possible to approximate the limit cycle stable manifold in situations where the limit cycle amplitude is much greater than the amount of freeplay.
Original languageEnglish
Pages (from-to)379-403
Number of pages25
JournalJournal of Applied Nonlinear Dynamics
Issue number2
Publication statusPublished - 1 Jun 2023


  • Limit cycle oscillations
  • Freeplay non-linearities
  • Floquet Analysis
  • Stability Domains
  • Aeroelasticity
  • Floquet analysis
  • Stability domains


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